POINCARÉ INEQUALITIES BASED ON BANACH FUNCTION SPACES ON METRIC MEASURE SPACES
MARCELINA MOCANU “Vasile Alecsandri” University of Bacău, Faculty of Engineering, Department of Power Engineering, Mechatronics and Computer Science, 157 Calea Mărăşeşti, Bacău, 600115, Romania, e-mail:mmocanu@ub.ro
We introduce a new type of first order Poincaré inequality for functions defined on a metric measure space, that is an useful tool in the study of Newtonian spaces based on Banach function spaces. This Poincaré inequality extends the Orlicz-Poincaré inequality introduced by Aïssaoui (2004) and the Poincaré inequality based on Lorentz spaces, introduced by Costea and Miranda (2011), that in turn generalize the well-known weak (1,p)-Poincaré inequality. Using very recent results of Durand-Cartagena, Jaramillo and Shanmugalingam (2012, 2013), it turns out that every complete metric space X, endowed with a doubling measure and supporting a weak Poincaré inequality based on a Banach function space is (thick) quasiconvex. We prove that the validity of the Poincaré inequality based on a Banach function space, on a doubling metric measure space, implies a pointwise estimate involving an appropriate maximal operator.
